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From: steve@cogsci.ed.ac.uk (Steve Finch)
Subject: Re: Fuzzy theory or probability theory?
Message-ID: <D01Hz9.FrG@cogsci.ed.ac.uk>
Organization: Centre for Cognitive Science, Edinburgh, UK
References: <3af1jc$3fg@nuscc.nus.sg> <3aqfdr$11b1@hearst.cac.psu.edu>   <1994Nov22.133236.2771@debet>,<3b02kn$jrc@b.stat.purdue.edu>   <3bdcgn$15jm@hearst.cac.psu.edu>,<D012oD.51F@cogsci.ed.ac.uk> <3bfc0f$taj@hearst.cac.psu.edu>
Date: Tue, 29 Nov 1994 17:30:40 GMT
Lines: 81
Xref: glinda.oz.cs.cmu.edu comp.ai.fuzzy:3482 sci.stat.math:3379

caj@jerry.psu.edu writes:

>In article <D012oD.51F@cogsci.ed.ac.uk>, steve@cogsci.ed.ac.uk (Steve Finch) writes:

>>
>>
>>> [ Oval pink shapes are almost red circles ]
>>
>>>Why would probability theory do better here since this is not a chance
>>>probability problem?  I am not concerned with the chances of finding a red
>>>and circular object.  I am concerned with how to classify this object that
>>>does not exactly fit my set description.  Making up a new set for each 
>>>different object is not practical in my system. And besides, such a method
>>>ignores the information that the object is very similar to an ideal
>>>member.
>>
>>>How would you classify this object while recognizing that it is very similar
>>>to the ideal member for this set?

>> [ Treat the problem with probability meaning the degree of plausibility 
>>   of "is describable by" ]


>No, here you are introducing trials (many people, a distribution of answers).
>And your use of prob theory after that works well.  But the question has no
>groups of people, no range of "possible" descriptions.  I simply want to
>conveniently and numerically encode that this object is not the same as the
>ideal reprsentative of the set but is very similar.  I am not interested in
>how many people would put it in the set and how many wouldn't.  *That* is
>a probability problem and handled quite well by Bayes.

I don't think you need to mention trials --- nothing in probability
theory depends on a trial based semantics, and Bayesianists positively
discourage it, preferring a personal "degree of plausibility/belief"
semantics.  I would say that it is crucial to fuzziness that there
*are* sets of "possible descriptions".  I just propose a different
method of assertaining the "degree of membership" of an object to a
description by redefining it to be "plausibility of description in
that context".

How can you encode in fuzzy theory the fact that a guppy is a good
"pet fish" without either being a good pet or a good fish?
Intuitively the notion of "degree of membership" is roughly "how many
prototypical properties apply?"  The right way to answer this question
is to describe these properties and look at probability distributions.

>Also, trying to predict that the object is a red circle given some description
>that uses the word oval is also a fine probability problem (as is most of
>detection and classification theory).  You propose to have lots of objects,
>some initial distribution of shapes, and some current sensor description.
>Unfortunately, you threw all that in on top of my problem.  Take that out
>and you can't use probability theory.  True?

True, but I would argue this is what fuzziness is all about --- we
know more information about things than we can (or would like to) put
in a description, and yet we have to describe it.  This involves
putting a quart of (mostly useless) information in a micro pint pot,
and information is lost.  Probability theory, not fuzzy set theory, is
the appropriate formalism to analyse this information loss, I think.

In this view, fuzzy theory is akin to a "hack" where "appropriateness
of description" is approximated.

>This also highlights what I tried to say before (maybe in another post).  You
>can have a fuzzy set and layer probability trials on it.  What's the chance
>of an object being a 0.7 member of set A (which might stand for being oval)?
>A=0.7 is simply an event and can have some chance of occuring. They work
>well together.

Indeed you can.  But does it really add anything?  Certainly not in
terms of formal power of the theory, methinks, since you can always
add functions satisfying the axioms of fuzzy set theory between
individuals and real numbers to your theory which stand as surrogate
degree-of-membership functions.  I guess you would want the event spae
to *be* these functions.

Cheers,

Steve.


